If it's just finding the derivative then this is probably more a question for the calculus thread.
Anyhoo, i'll do the first part. Take logs to get.
ln(y)= 5x ln(4-x)
Now differentiate (implicitly on the LHS) and rearrange to get your answer for dy/dx. Have a go and let me know...
I guess that depends what the question is. Are you asking for the differential equation to which this is the solution?
Take logs and differentiate for first order ODE?
Yeah that amused me too :smile: .I'd probably start a little earlier:
Growth proportional to size implies.....
\frac{dy}{dt}=ky
with the intial condition y(0)=500 and the time t = 3 (in hours!) condition y(3)=8000.
Your mission, should you choose to accept it, is to find the general...
and (b) gives:
(.....i'll spare the details but it's the usual integrating factor prob...)
y = x(cx - 3)
where c is your integration constant.
To satisfy the IC y(0) = 0, c can take on any value you want.
Just to put this problem in a general context, it's form is:
\frac {dy} {dx} = a(x) y + b(x) y^p
Which is a Bernoulli ODE (or a Ricatti with no constant term).
The substitution:
u(x) = y^{1-p}
reduces this to a first order linear ODE which can be solved in the usual way via an...